2017 AMC 12A 第 4 题

Question

Jerry 和 Silvia 想从一块正方形田地的西南角走到东北角。Jerry 先向正东走，再向正北走到达目的地；Silvia 则朝东北方向沿直线走到目的地。与 Jerry 的路程相比，Silvia 的路程大约短了多少？

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia’s trip was, compared to Jerry’s trip?

30\%

40\%

50\%

60\%

70\%

答案：A

知识点：勾股定理百分数

难度评级：1200

小提示：

设边长为

x

；Jerry 走

2x

，Silvia 走对角线

Let the side be

x

; Jerry walks

2x

and Silvia walks the diagonal

大提示：

对角线为

x\sqrt2

，节省的比例是

\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2}

。

The diagonal is

x\sqrt2,

so the fraction saved is

\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2}

解答：

如果正方形边长为

x

，Jerry 走

x+x=2x

而 Silvia 走对角线

\sqrt{x^2+x^2}=x\sqrt2

。

Silvia 的路程短了的比例为

\begin{aligned} &\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2} \\ &\approx 1-0.707=0.293 \end{aligned}\text{。}

这最接近

30\%

。

所以正确答案是 A。

If the square has side

x,

Jerry walks

x+x=2x,

while Silvia walks the diagonal

\sqrt{x^2+x^2}=x\sqrt2.

The fraction by which Silvia’s trip is shorter is

\begin{aligned} &\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2} \\ &\approx 1-0.707=0.293. \end{aligned}

This is closest to

30\%.

Thus, the correct answer is A.

2017 AMC 12A 第 4 题

答案：A

难度评级：1200

小提示：

大提示：

解答：

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